Preferences

Risk Aversion

• To model liking high expected returns and disliking high variance, we can rank portfolios based on a combination of mean and variance:

Indifference curves

Benefit-cost ratio at the optimum

• For frontier portfolios, the marginal benefit-cost ratio is the same across assets:
  • marginal benefit / marginal cost \(= k\)
  • marginal benefit = risk premium
  • marginal benefit is the marginal contribution to portfolio risk \(C_i^\top w\)
  • \((\bar{r}_i-r_f)/C_i^\top w = k\)
• The optimal frontier portfolio for risk aversion \(A\) is the one for which the marginal benefit-cost ratio equals \(A\).

Example

import numpy as np

rf = 0.02
mn1, mn2, mn3 = 0.06, 0.08, 0.10
sd1, sd2, sd3 = 0.1, 0.15, 0.12
corr12, corr13, corr23 = 0.5, 0.7, 0.6

raver = 6

Solution of the example

S = np.diag([sd1, sd2, sd3])
R = np.identity(3)
R[0, 1] = R[1, 0] = corr12
R[0, 2] = R[2, 0] = corr13
R[1, 2] = R[2, 1] = corr23
C = S @ R @ S

rprem = np.array([mn1, mn2, mn3]) - rf
Cinv = np.linalg.inv(C)
w = (1/A) * Cinv @ rprem

w1=-13.3%, w2=1.1%, w3=62.5%